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Rheology of Frictional Grains Rheology of frictional grains Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universität Göttingen im Promotionsprogramm ProPhys der Georg-August University School of Science (GAUSS) vorgelegt von Matthias Grob aus Mühlhausen Göttingen, 2016 Betreuungsausschuss: • Prof. Dr. Annette Zippelius, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Dr. Claus Heussinger, Institut für Theoretische Physik, Georg-August-Universität Göttingen Mitglieder der Prüfungskommission: • Referentin: Prof. Dr. Annette Zippelius, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Korreferent: Prof. Dr. Reiner Kree, Institut für Theoretische Physik, Georg-August-Universität Göttingen Weitere Mitglieder der Prüfungskommission: • PD Dr. Timo Aspelmeier, Institut für Mathematische Stochastik, Georg-August-Universität Göttingen • Prof. Dr. Stephan Herminghaus, Dynamik Komplexer Fluide, Max-Planck-Institut für Dynamik und Selbstorganisation • Dr. Claus Heussinger, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Prof. Cynthia A. Volkert, PhD, Institut für Materialphysik, Georg-August-Universität Göttingen Tag der mündlichen Prüfung: 09.08.2016 Zusammenfassung Diese Arbeit behandelt die Beschreibung des Fließens und des Blockierens von granularer Materie. Granulare Materie kann einen Verfestigungsübergang durch- laufen. Dieser wird Jamming genannt und ist maßgeblich durch vorliegende Spannungen sowie die Packungsdichte der Körner, welche das Granulat bilden, bestimmt. Die Rheologie dichter granularer Medien ist zusätzlich zu Spannung und Packungsdichte stark durch Reibung zwischen den Körnern beeinflusst. Wir zeigen mittels numerischer Simulationen und analytischer Betrachtungen, wie Reibung Jamming qualitativ verändert. Reibungsfreies Jamming ist ein kon- tinuierlicher Phasenübergang mit einem kritischen Punkt bei verschwindender Spannung. Reibungsbehaftetes Jamming ist ein diskontinuierlicher Phasenüber- gang mit einem kritischen Punkt bei endlicher Spannung. Der kritische Punkt bei endlicher Spannung führt zu bemerkenswertem Verhalten: Oberhalb der kri- tischen Packungsdichte gibt es ein Intervall an Packungsdichten, innerhalb dessen große oder kleine Spannungen zum Fließen führen, mittlere Spannungen hingegen führen zum Blockieren des Mediums. Das Fließverhalten nahe Jamming ist stark durch die Systemgröße beeinflusst: Es gibt eine kritische Systemgröße, oberhalb derer zeitabhängiger Fluss entsteht. Dieser zeitabhängige Fluss wird durch die Ausbildung von großskaligen Strukturen im Spannungsfeld erklärt. Sowohl die großskaligen Strukuren als auch der damit einhergehende zeitabhängige Fluss sind neuartige Phänomene im Fluss von trockenen Granulaten und durch Rei- bung hervorgerufen. iii Abstract This thesis deals with the description of flow and arrest of granular matter. Granular matter can undergo a rigidity transition — called jamming — that is mainly controlled by the applied stresses and the packing fraction of the grains that constitute the medium. In addition to stress and packing fraction, inter- particle friction greatly affects the rheology of granular matter. Using numerical simulations and analytical modeling, we show how novel behavior in dense flow and jamming regimes arises in the presence of friction. In particular, frictionless jamming is continuous with a critical point at zero stress. In contrast, frictional jamming is shown to exhibit a discontinuous phase transition with a critical point at finite stress. The fact that the critical point resides at finite stress gives rise to remarkable flow behavior, called reentrant flow. Explicitly, there is an interval of packing fractions above the critical packing fraction in which large or low stress leads to flow but intermediate stress jams the medium. The behavior close to jamming depends substantially on the system size, i.e., there is a critical sys- tem size above which unsteady flow emerges. Unsteady flow is rationalized by large-scale structures in the stress fields. Both, the large-scale structures and the accompanied unsteady flow, are novel phenomena regarding the flow of dry granular matter and can be attributed to interparticle friction. v Contents I. Introduction 1 1. An introduction to granular media 3 2. Scope of the thesis 5 3. Models for grains and friction 7 3.1. Deformation of frictionless spheres . .7 3.2. Frictional interactions . 10 4. The jamming transition 13 4.1. Frictionless jamming . 15 4.2. Frictional jamming . 18 5. Shear rheology 21 5.1. Rheometry and flow curves — an overview . 21 5.2. Inertial flow . 24 5.3. Plastic flow . 25 5.4. Shear thickening . 27 5.5. Unsteady flow . 30 6. Active microrheology 33 II. Results 35 7. Shear rheology of frictional grains 39 7.1. Jamming of frictional particles . 39 vii Contents 7.2. Rheological chaos of frictional grains . 46 7.3. Unsteady rheology and heterogeneous flow of dry frictional grains 52 8. Active microrheology of driven granular particles 65 III. Summary and discussion 73 IV. Outlook 79 V. Appendix 85 A. Details on the hydrodynamic model 87 A.1. The flowing state . 88 A.2. The shear jammed state . 89 Bibliography 91 Author contribution 105 Acknowledgments 107 List of Publications 109 viii Part I. Introduction 1 1. An introduction to granular media If we measure it by tons, the material most manipulated by man is water; the second-most-manipulated is granular matter. (de Gennes [1999]) Granular media — collections of grains — are ubiquitous in nature, daily life, and engineering disciplines. Grains constitute the seabed, dunes, beaches, and an important proportion of the earth’s soil. We use pepper, sugar, and salt, coffee beans or powder, and many other kinds of granular media in the kitchen and in our daily life. Sand is a resource of which glass, semiconductors, concrete, and ultimately buildings are made. Granular matter as a resource or consumer products is handled in industrial processes, which, in total, are estimated to be responsible for a tenth of the planet’s energy budget [Mullin, 2002]. The amount of energy for the handling of grains, i.e., for processes where granular media are deformed or transported, highlights the importance of a proper understanding of flow properties of granular matter as a complex fluid. Grains are composed of a vast number of molecules and the energy associated with motion or contact is by orders of magnitude larger than the thermal en- ergy, kBT . Granular media are therefore considered athermal [Bi et al., 2015; Brilliantov and Pöschel, 2010]. Interactions between grains are inelastic, i.e., in a collision, energy is dissipated into internal degrees of freedom, see, e.g., [Lan- dau and Lifshitz, 1970]. Furthermore, when grains are in contact, any motion or force parallel to the contact surface is opposed by a frictional force [Johnson, 1985]. Energy scale separation, the inelastic character of collisions, and friction make granular media distinct; classical equilibrium thermodynamics and fluid dynamics are not applicable [Kadanoff, 1999]. 3 1. An introduction to granular media Granular media are reminiscent of fluids or solids but exhibit a host of unusual phenomena [Jaeger et al., 1996]. In contrast to ordinary (simple) fluids, granular fluids do not possess a constant viscosity, i.e., the ratio between shear stress and strain rate depends on the applied shear. An excessive increase in viscosity can ultimately lead to a transition into a solid disordered state with a finite shear modulus. The transition from fluid to solid is called jamming. The solid state, too, shows extraordinary behavior. Granular particles are not bound to each other by chemical bonds but by confinement in a volume. An infinitesimal extension of the confining volume can lead to the complete loss of a finite shear modulus and the initiation of a flowing state. Rheology studies flow problems of fluids which are non-Newtonian, i.e., fluids that possess a viscosity that is not constant as a function of strain rate or shear stress. The rheological properties depend on a number of aspects and parameters: the packing fraction of the particles, the particles themselves and their interaction, the geometry and boundaries of the problem, and the preparation history. In this thesis, we study these aspects, with a particular focus on the exceptional role played by friction, in a granular medium under shear. 4 2. Scope of the thesis In the course of this thesis we deal with dense granular media under shear. In part I, we give an overview of the subject. In Chapter 3, we deal with molecular models of granular particles. Chapter 4 focuses on assemblies of grains and explains the jamming transition: a rigidity transition from a flowing state to a disordered solid state. Subsequently, Chapter 5, deals with the shear flow of densely packed particles including phenomena like shear thickening and flow heterogeneities. Chapter 6 describes an approach to probe a complex fluid by the dynamics of a probe particle. In part II, we present the results of the thesis. This part is divided in two Chapters. Chapter 7 constitutes the main results of this thesis and we investi- gate the rheology of frictional grains. This Chapter is divided in three sections corresponding to an article each. First, in section 7.1, we discuss the jamming transition of frictional grains in small simulation cells. Results on large systems, in which an unsteady and chaotic response emerges, are presented in section 7.2. There, we also present a simple
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